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\begin{document}
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\title{A Lyapunov-based Solution for Precise Auto-landing of UAVs on Mobile Platforms}

\author{
\name{Darmesh Kumar\textsuperscript{a}, Jai Raj\textsuperscript{a}, Krishna Raghuwaiya\textsuperscript{b} and Jito Vanualailai\textsuperscript{a}\thanks{CONTACT Jai Raj. Email: jai.raj@usp.ac.fj}}
\affil{\textsuperscript{a}School of Information Technology, Engineering, Mathematics \& Physics, Laucala Campus, The University of the South Pacific, Suva, Fiji; \textsuperscript{b}School of Pacific Arts, Communication and Education, Laucala Campus, The University of the South Pacific, Suva, Fiji}
}

\maketitle
\begin{abstract}
In this paper, we present an autonomous system for landing point-mass robots on car-like mobile landing platforms in 3-dimensional space using Lyapunov-based control scheme (LbCS). Herein, we have designed the nonlinear, time-invariant acceleration-based control algorithm for the Autonomous Vehicles (AVs), which was extracted using the LbCS. Fixed and moving obstacles were included in the dynamic environment under consideration, which the point-mass robots and the mobile landing platforms avoided while in motion to their respective targets. The control laws take into consideration cylindrical obstacles that mimic buildings in a city-like environment. In addition, the Minimum Distance Technique (MDT) has been used to derive the mathematical functions related to cylindrical obstacles and their successful avoidance. To demonstrate the stability of the dynamic model, the direct Lyapunov approach is utilized. Computer simulations of real-world situations were used to demonstrate the effectiveness, robustness, and advantages of the control scheme. Future work will involve experimentally evaluating this strategy on an actual prototype.

\end{abstract}

\begin{keywords}
Lyapunov; Point-mass robots; Car-like mobile robots; Landing Platform; Safe landing; Stability
\end{keywords}

\section{Introduction}
Over the past few decades, researchers have been considerably focusing on the development of Autonomous Vehicles (AVs), namely Unmannned Aerial Vehicles (UAVs), which progressively influences humans and their lives, due to their advantages over human-piloted aircrafts with applications in manufacturing, industrial, civillian and military  sectors \cite{jairajIEEEAccess}. Miniature UAVs are increasingly developing in popularity but are still within their earliest stage of development in terms of its mass appropriation and utilisation \cite{Cai2010}. It is very beneficial for quick conveyances at peak hours and also amazingly advantageous in areas where humans find it difficult to reach or are incapable to perform in a convenient and proficient way \cite{Cai2010, jairajIEEEAccess, raj2023autonomous}. These tasks could be classified as the 3D's of robotic applications, namely dirty, dull and dangerous \cite{jairaj2018}. Various applications also require multiple UAVs. Thus the navigation and control of AVs with the involvement of multiple coordinated and connected mobile aerial vehicles, play a vital role in many real-life applications, such as military-based applications, search and rescue, disaster relief, agriculture automation, video shooting, maintenance, industrial, surveillance and transportation, to name a few \cite{Cai2010, kalantari2017, maziar2017, jairaj2018, raghuwaiya2018, rios2019, jairajHINDAWI2020, jairajIEEEAccess, chand2020, jairajROBOTICA2021, raj2023autonomous}.

Since the recent COVID-19 outbreak which originated from Wuhan, China in December 2019, the government and authorities all over the globe are struggling to control the spread of this virus \cite{FONG2020covid, jat2020covid}.  During the COVID-19 pandemic, many countries have deployed UAVs for various purposes. These include surveillance of their containment areas, delivery and transportation of medical supplies, disinfecting and sterilising public spaces, detecting body temperatures using thermal detectors, which provides security and comfort to frontline workers by reducing human contact and potential spread of the virus \cite{zeng2020covid, jain2021covid}. Micromulticopter drones were used in Shenzhen city in China to transport medical samples and perform thermal imaging \cite{covid_drones_shenzhen_2020}. In other applications, drones in Dubai were used to produce fake rains by releasing electrical charges into the clouds, causing a downpour\cite{dubai_fake_rain_2021}. This method is known as ‘cloud-seeding’ where the clouds have to come together to allow precipitation and produce the rain. In smart cities, UAVs are used in the transportation sector to provide solutions to manage traffic congestions, measure air pollution and conveyances, to name a few \cite{jairajIEEEAccess, batty2012smartcity, chourabi2012smartcity}.

Furthermore, in the motion planning and control problem, the connectedness of AVs are important. The vehicle-to-vehicle (V2V), vehicle-to-infrastructure (V2I) and vehicle-to-everything (V2X) communications use wireless and cellular networks to govern their connectedness \cite{raghuwaiya2018, jairajROBOTICA2021, raj2023autonomous}. These V2V, V2I and V2X collaborations have significant advantages linked to improving traffic management, providing driver assistance, improving fuel efficiency, direction and route optimisation and preventing possible accidents. These advantages are associated with the transportation sector in a \emph{smart-city} \cite{festag2014, hamida2015}. The focus of this research paper is with UAVs and Unmanned Ground Vehicles (UGVs). The fundamental problem of any robotic system is to be able to perform collision-free navigation in environments cluttered with both static and dynamic obstacles. The environment in which the AVs maneuver is either prior-known or unknown, as such, it should be equipped with navigation systems that are automated to ensure collision-free navigation \cite{jairaj2016, jairaj2018APWC}. A key issue for most real-life applications of AVs robots is to be able to perform safe and collision-free navigation in the presence of obstacles. Over the decades, even with extensive research, this fundamental problem has remained at large a real challenge for researchers \cite{jairaj2018, raghuwaiya2018}.

\subsection{Contributions}
This paper, which builds on \cite{darmesh2021}, will focus on a multiple vehicular system. The UAVs are modelled as point-mass mobile robots and the landing platforms are modelled as car-like mobile robots. Each point-mass mobile robot will land on a unique mobile landing platform. The vehicular systems will perform collision-free navigation and obstacle avoidance maneuvers in a constrained environment. Cylindrical obstacles will be introduced to mimic buildings in a smart-city like environment \cite{jairajJV2021, jairajJR2021, jairajKSR2021}. To avoid these cylindrical obstacles, the Minimum Distance Technique (MDT) which was introduced in \cite{sharma2012mdt, sharma2014mdt} will be utilized. The MDT calculates the minimum distance from the center of the mobile robot to the closest point on the curved surface wall of the cylinder. The avoidance of the closest point on the curved surface wall of the cylinder intuitively guarantees the avoidance of the entire curved surface wall of the cylinder. The Lyapunov-based Control Scheme (LbCS), which is an Artificial Potential Field (APF) method, is an effective technique to derive the acceleration-based control laws to ensure safe navigation within a $3$-dimensional workspace \cite{jairaj2018}. The LbCS will not only provide smooth trajectories but also allows for an easier construction of mathematical functions from limitations, inequalities, restrictions and mechanical constraints, and incorporates them into the control laws \cite{jairaj2018, sharma_vanualailai_singh_2017, sharma2009flocking}.

The main contributions of this paper are summarised as follows:
\begin{enumerate}
  \item The design of the nonlinear, time-invariant acceleration-based control algorithm for AVs based on a Lyapunov function that acts as an energy function.
  \item A dynamic environment of AVs includes multiple point-mass mobile robots performing autonomous landing on car-like mobile platforms while avoiding static and dynamic obstacles.
  \item A $3$-dimensional rectangular workspace whose boundaries are treated as fixed obstacles. The robotic system utilizes the V2V, V2I and V2X communications with the workspace for safe robot navigation.
\end{enumerate}

The paper is outlined as follows: Section \ref{section_robot_model_evric} represents the system architecture of our robotic system; Section \ref{LbCS_evric} defines the APF functions; in Section \ref{NLCD_evric}, the Lyapunov function is constructed and the nonlinear acceleration-based control laws are extracted; Section \ref{stability_evric} presents the stability analysis of the robotic system; in Section \ref{section_sim_evric}, the simulation results are illustrated to demonstrate the effectiveness and robustness of the control scheme; and finally, Section \ref{section_conclusion_evric} concludes the paper with future work.

\section{Mathematical Model of the System}\label{section_robot_model_evric}
Mathematical modelling is a process of translating between real world and mathematics. It is a process whereby models, representing real-life scenarios, are developed by carefully noting down assumptions which include the restrictions, limitations and the dynamical constraints of a system. It helps to explain the system and study the effects of various components, and make a prediction about the behavior of the real system. These models are then solved and the solutions are interpreted in terms of real-life problems. This section deals with the kinematics and dynamics of a system of multiple UAVs, modelled as point-masses, and car-like mobile landing platforms.

\subsection{UAV Model}
This section proposes a simple kinematic model of a UAV modelled as a point-mass. Figure \ref{fig:UAV_pointmass_model_evric} shows the schematic representation of the UAV model in $2$-dimensional space with an obstacle avoidance scenario.
%%----------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{figure*}
\centering
\includegraphics[width=0.6\textwidth]{UAV_pointmass_model_evric.pdf}
\caption{A representation of the kinematic model of UAVs in $2$-dimensional space.}
\label{fig:UAV_pointmass_model_evric}
\end{figure*}
%%-----------------------------------------------------------------------------------------------------------------------------------------------------------
Thus, in a $3$-dimensional workspace, we can define the $i$th UAV as follows:
\begin{definition}
The $i$th UAV modelled as a point-mass, ${\mathcal{P}_i}$, is a sphere of radius ${r{p_i}}$ and centred at $\left( {x{p_i}\left( t \right),y{p_i}\left( t \right),z{p_i}\left( t \right)} \right) \in {\mathbb{R}^3}$ for every time $t \geqslant 0$. The $i$th point-mass is the set:
\begin{equation*}
  \mathcal{P}_i = \left\{ \begin{gathered}
  \left( {{Z_1},{Z_2},{Z_3}} \right) \in {\mathbb{R}^3}: {\left( {{Z_1} - x{p_i}} \right)^2} + {\left( {{Z_2} - y{p_i}} \right)^2} + {\left( {{Z_3} - z{p_i}} \right)^2} \leqslant r{{p_i}^2}
\end{gathered}  \right\},
\end{equation*}
for $i=1,\ldots,n$.
\end{definition}

At time $t \geqslant 0$, we let the instantaneous velocities of ${\mathcal{P}_i}$ to be  \(\left( {\nu {p_i}\left( t \right),\omega {p_i}\left( t \right),\mu {p_i}\left( t \right)} \right) = \left( {\dot x{p_i}\left( t \right),\dot y{p_i}\left( t \right),\dot z{p_i}\left( t \right)} \right)\) and the instantaneous accelerations to be $\left( {{\sigma _{i1}}\left( t \right),{\sigma _{i2}}\left( t \right),{\sigma _{i3}}\left( t \right)} \right) = \left( {\dot \nu {p_i}\left( t \right),\dot \omega {p_i}\left( t \right),\dot \mu {p_i}\left( t \right)} \right)$. Assuming the initial conditions at $t = {t_0} \geqslant 0$, a system of first order ODE's of ${\mathcal{P}_i}$ is:

\begin{equation}\label{UAV_pointmass_ODE_evric}
 \left. \begin{gathered}
  \dot x{p_i} = \nu {p_i},\dot y{p_i} = \omega {p_i},\dot z{p_i} = \mu {p_i}, \hfill \\
  \dot \nu {p_i} = {\sigma _{i1}},\dot \omega {p_i} = {\sigma _{i2}},\dot \mu {p_i} = {\sigma _{i3}}, \hfill \\
  x{p_{i0}} = x{p_i}\left( {{t_0}} \right),y{p_{i0}} = y{p_i}\left( {{t_0}} \right),z{p_{i0}} = z{p_i}\left( {{t_0}} \right), \hfill \\
  \nu {p_{i0}} = \nu {p_i}\left( {{t_0}} \right),\omega {p_{i0}} = \omega {p_i}\left( {{t_0}} \right),\mu {p_{i0}} = \mu {p_i}\left( {{t_0}} \right). \hfill \\
\end{gathered}  \right\}
\end{equation}

\subsection{Landing Platform Model}
We consider a rear wheel driven car-like mobile robot in $3$-dimensional workspace whose engine power is applied to the rear wheel. For easier representation, a $2$-dimensional model of the car-like mobile robot is designed as shown in Figure \ref{fig:vehicle_model_evric} with a reference point of $\left( x{v_i},y{v_i}\right)$. Let $\theta _i$ represent the vehicle's orientation with respect to $Z_1$-axis and $\phi {v_i}$ be the steering angle of the vehicle.
%%----------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{figure*}
\centering
\includegraphics[width=0.4\textwidth]{vehicle_model_evric.pdf}
\caption{Kinematic model of a rear wheel driven car-like robot with rear wheel steering and steering angle $\phi {v_i}$.}
\label{fig:vehicle_model_evric}
\end{figure*}
%%-----------------------------------------------------------------------------------------------------------------------------------------------------------
Thus, the $i$th car-like mobile landing platform can be defined as follows:
\begin{definition}
The $i$th mobile landing platform, ${\mathcal{V}_i}$, is modelled as a car-like robot which is enclosed by a protective spherical region centred at $\left( {x{v_i}\left( t \right),y{v_i}\left( t \right),z{v_i}\left( t \right)} \right) \in {\mathbb{R}^3}$ with radius $r{v_i}: = \frac{1}{2}\sqrt {{{(2{\varepsilon _1} + L)}^2} + {{(2{\varepsilon _2} + l)}^2}}$, given the length $L$, the width $l$ and the \emph{clearance parameters} $\varepsilon _1$ and $\varepsilon _2$, for $t \geqslant 0.$ Precisely, it is a set:
\begin{equation*}
\mathcal{V}_i = \left\{ \begin{gathered}
  \left( {{Z_1},{Z_2},{Z_3}} \right) \in {\mathbb{R}^3}: {\left( {{Z_1} - x{v_i}} \right)^2} + {\left( {{Z_2} - y{v_i}} \right)^2} + {\left( {{Z_3} - z{v_i}} \right)^2} \leqslant r{v_i}^2
\end{gathered}  \right\},
\end{equation*}
for $i=1,\ldots,n$.
\end{definition}

At time $t \geqslant 0$, we let the instantaneous velocities of ${\mathcal{V}_i}$ to be $\left( {\nu {v_i}\left( t \right),\omega {v_i}\left( t \right),\mu {v_i}\left( t \right)} \right)$ and the instantaneous acceleration to be $\left( {{\sigma _{i4}}\left( t \right),{\sigma _{i5}}\left( t \right),{\sigma _{i6}}\left( t \right)} \right) = \left( {\dot \nu {v_i}\left( t \right),\dot \omega {v_i}\left( t \right),\dot \mu {v_i}\left( t \right)} \right)$. Thus, if we allow the mass of the $i$th vehicle to be $m_i$, $F_i$ to be the force along the axis of the vehicle, $\daleth_i$ the torque about the vertical axis at $\left( x{v_i},y{v_i}\right)$ and $I_i$ to represent the moment of inertia of the vehicle, then the system of first-order ODE's is derived as:
\begin{equation}\label{vehicular_ODE_evric}
\left. \begin{gathered}
  \dot x{v_i} = \nu {v_i}\,\cos \,{\phi {v_i}} - \frac{L}{2}\omega {v_i}\,\sin\, {\phi {v_i}}, \hfill \\
  \dot y{v_i} = \nu {v_i}\,\sin\, {\phi {v_i}} + \frac{L}{2}\omega {v_i}\,\cos\,{\phi {v_i}} , \hfill \\
  \dot z{v_i} = \mu {v_i}, \hfill \\
  \dot \phi {v_i} := \omega {v_i},\hfill\\
  \dot \nu {v_i} := {\sigma _{i4}} = \frac{F_i}{m_i},\hfill\\
  \dot \omega {v_i} := {\sigma _{i5}} = \frac{\daleth _i }{I_i},\hfill\\
  \dot \mu {v_i} := {\sigma _{i6}} \hfill \\
\end{gathered}  \right\}
\end{equation}

\subsection{Dynamic Model of the Team}
Dynamical model of mobile robots is very essential in situations where the velocities of the mobile robots generate forces whose significance cannot be ignored. Since kinematic models are restricted to low velocity, low acceleration and limited loads, dynamical model become useful for mobile robots travelling at higher velocities and carries heavy payloads.With reference to the systems (\ref{UAV_pointmass_ODE_evric}) and (\ref{vehicular_ODE_evric}), the combined dynamic model of the team of ${\mathcal{P}_i}$ and ${\mathcal{V}_i}$ are as follows:
\begin{equation}\label{first_order_ODE_evric}
 \left. \begin{gathered}
  \dot x{p_i} = \nu {p_i},\dot y{p_i} = \omega {p_i},\dot z{p_i} = \mu {p_i}, \hfill \\
  \dot \nu {p_i} := {\sigma _{i1}},\dot \omega {p_i} := {\sigma _{i2}},\dot \mu {p_i} := {\sigma _{i3}}, \hfill \\
  \dot x{v_i} = \nu {v_i}\, \cos\, {\phi {v_i}}  - \frac{L}{2}\omega {v_i}\, \sin\, {\phi {v_i}}, \hfill \\
  \dot y{v_i} = \nu {v_i}\, \sin\,{\phi {v_i}}  + \frac{L}{2}\omega {v_i}\, \cos\, {\phi {v_i}}, \hfill \\
  \dot z{v_i} = \mu {v_i}, \dot \phi {v_i} := \omega {v_i},\hfill\\
  \dot \nu {v_i} := {\sigma _{i4}} = \frac{F_i}{m_i},\hfill\\
  \dot \omega {v_i} := {\sigma _{i5}} = \frac{\daleth _i }{I_i},\hfill\\
  \dot \mu {v_i} := {\sigma _{i6}} \hfill \\
\end{gathered}  \right\}
\end{equation}
for $i=1,\ldots,n$.
%.............................................................................................................................................................................
\section{Lyapunov-based Control Scheme}\label{LbCS_evric}
This section will illustrate the work of LbCS (an artificial potential field method), which when the attraction and repulsive functions are combined, forms the total energy function of the system and generate a force to minimize the energy to a minimum value at the goal configuration. The artificial force induced by the sum of the attractive and repulsive forces will generate the motion direction the robots need to follow \cite{minguez2002}. The main aim is to utilize the artificial potential field function to design the translational and rotational accelerations that will ensure the robots to safely navigate in the cluttered environment and converge to their designated targets in a $3$-dimensional workspace.

The attractive and repulsive functions will combine to control the motion of the robots by forcing them away from obstacles and attracting towards their designated targets.
\subsection{An Example of an Artificial Potential Field Function}
The work of LbCS is illustrated in this section with a simple case where a point-mass robot with radius $r_p$ moves to its target while avoiding all stationary spherical obstacles. For time $t \geqslant 0$, the Euclidean distance between the point-mass robot and the target needs to be calculated for target attraction. For simplicity, a $2$-dimensional workspace is considered.
\begin{definition}
A point-mass robot has a disk-shaped target, $\mathcal{T}$, with center $\left(\tau_1,\tau_2\right)$ and radius $r_\tau$. It is the set:
\begin{equation*}
 \mathcal{T} = {\text{ }}\left\{ {\left( {{Z_1},{Z_2}} \right) \in {\mathbb{R}^2}:{{\left( {{Z_1} - {\tau _1}} \right)}^2} + {{\left( {{Z_2} - {\tau _2}} \right)}^2} \leqslant {r_\tau} ^2} \right\}.
\end{equation*}
\end{definition}
Thus the attractive potential function is
\begin{equation*}
 V\left( \mathbf{x} \right) := \frac{1}{2}\left[ {{{\left( {x - {\tau _1}} \right)}^2} + {{\left( {y - {\tau _2}} \right)}^2}} \right].
\end{equation*}
The minimum Euclidean distance between the point-mass robot and the fixed obstacle also needs to be measured for obstacle avoidance. Consider $m \in \mathbb{N}$ to be the number of obstacles fixed in the workspace.
\begin{definition}
The $k$th stationary object is a disk with center $\left( {{o_{k1}},{o_{k2}}} \right)$ and radius ${r_o}$. It is the set:
\begin{equation*}
 \mathcal{O} = \left\{ {\left( {{Z_1},{Z_2}} \right) \in {\mathbb{R}^2}:{{\left( {{Z_1} - {o_{k1}}} \right)}^2} + {{\left( {{Z_2} - {o_{k2}}} \right)}^2} \leqslant {ro_k}^2} \right\}
\end{equation*}
\end{definition}
Thus the obstacle avoidance function is
\begin{equation*}
  FO\left( \mathbf{x} \right) := \frac{1}{2}\left[ {{{\left( {x - {o_{k1}}} \right)}^2} + {{\left( {y - {o_{k2}}} \right)}^2} - {{\left( {{ro_k} + {rp}} \right)}^2}} \right]
\end{equation*}
for $k=1,\ldots,m$.

When the obstacle avoidance function (placed in the denominator) is suitably combined with a positive constant in the numerator, it forms the repulsive potential fields. Thus, the LbCS requires all obstacle avoidance functions to be combined with tuning parameters, and the capability of the obstacle avoidance depends on the formation of the repulsive potential fields.

Consider the following Lyapunov function with a tuning parameter $\beta > 0$.
\begin{equation}\label{Lyapunov_eg_evric}
  L\left( \mathbf{x} \right) = V\left( \mathbf{x} \right) + V\left( \mathbf{x} \right)\frac{\beta }{{FO\left( \mathbf{x} \right)}}
\end{equation}

As an illustration, we consider a scenario of $5$ obstacles positioned in a workspace with centers $\left( {{o_{11}},{o_{12}}} \right) = \left( {4,15} \right),\left( {{o_{21}},{o_{22}}} \right) = \left( {9,18} \right),\left( {{o_{31}},{o_{32}}} \right) = \left( {12,14} \right),\left( {{o_{41}},{o_{42}}} \right) = \left( {10,8} \right),\left( {{o_{51}},{o_{52}}} \right) = \left( {17,9} \right)$
and corresponding radius of ${ro_1} = 2.5,{ro_2} = {ro_3} = 1.5,{ro_4} = 3,{ro_5} = 2$. Let the initial position of a point-mass robot be $\left( {{p_1},{p_2}} \right) = \left( {1,19} \right)$ with radius $rp = 0.2$ and the target having a of center $\left( {{\tau _1},{\tau _2}} \right) = \left( {19,2} \right)$ with radius $r_\tau = 0.2$. Choosing the control parameter to be $\beta = 8$, the potential field generated by the attractive function and the repulsive potential created by $V\frac{\beta }{{FO}}$, is shown in Figure \ref{fig:Total_potentials_evric}.
%%-----------------------------------------------------------------------------
\begin{figure*}
$
\centering
\begin{array}{cc}
{\subfloat[3D visualization.]{\scalebox{0.47}{\includegraphics[width=1.0\textwidth]{Total_potential_3D.pdf}\label{total_potential_3D_evric}}}}&
{\subfloat[Contour plot.]{\scalebox{0.47}{\includegraphics[width=1.0\textwidth]{Total_potential_contourplot2.pdf}\label{total_potential_contourplot_evric}}}}
\end{array}
$
\caption{The total potentials generated using the Lyapunov function $L$ governed by Equation (\ref{Lyapunov_eg_evric}) for target attraction and obstacle avoidance, and the corresponding contour plot..}
\label{fig:Total_potentials_evric}
\end{figure*}
%%-----------------------------------------------------------------------------
Similarly, the above approach will be used to design the attractive and repulsive potential functions that will govern the motion of the UAVs and the car-like mobile platforms in a dynamiv environment.

\subsection{Target Attraction Functions}
A target is fixed for every mobile robot. Thus, for each mobile robot to get attracted to its target, the target attraction functions are formulated for $\mathcal{P}_i$ and $\mathcal{V}_i$, for $i=1,\ldots,n$.
\subsubsection{Attraction function for $\mathcal{P}_i$}
\begin{definition}
The target, $\mathcal{T}p_i$, of a UAV point-mass robot, ${\mathcal{P}_i}$, is a mobile car-like robot, $\mathcal{V}_i$, enclosed in a protective spherical region with centre $\left( {x{v_i},y{v_i},z{v_i}} \right)$  and radius $r{v_i}$. It is the set:
\begin{equation*}
  \mathcal{T}{p_i} = \left\{ \begin{gathered}
  \left( {{Z_1},{Z_2},{Z_3}} \right) \in {\mathbb{R}^3}: {\left( {{Z_1} - x{v_i}} \right)^2} + {\left( {{Z_2} - y{v_i}} \right)^2} + {\left( {{Z_3} - z{v_i}} \right)^2} \leqslant r{{v_i}^2}
\end{gathered}  \right\},
\end{equation*}
for $i=1,\ldots,n$.
\end{definition}

To ensure the convergence of each ${\mathcal{P}_i}$ to its mobile target ${\mathcal{V}_i}$ in the $3$-dimensional workspace, the target attraction function is
\begin{equation}\label{Attraction_for_UAV_pointmass_evric}
V{p_i} = \frac{1}{2}\left[ \begin{array}{l}{\left( {x{p_i} - x{v_i}} \right)^2} + {\left( {y{p_i} - y{v_i}} \right)^2} + {\left( {z{p_i} - \left( {z{v_i} + r{v_i} + r{p_i}} \right)} \right)^2}\\
 + \nu {p_i}^2 + \omega {p_i}^2 + \mu {p_i}^2
\end{array} \right]
\end{equation}
for $i=1,\ldots,n$.
%
\subsubsection{Attraction function for ${\mathcal{V}_i}$}
\begin{definition}
The target, $\mathcal{T}v_i$, of a car-like mobile landing platform, ${\mathcal{V}_i}$, is a fixed sphere with centre $\left( {\tau _{i1},\tau _{i2},\tau _{i3}} \right)$  and radius $r{\tau_i}$. It is the set:
\begin{equation*}
  \mathcal{T}{v_i} = \left\{ \begin{gathered}
  \left( {{Z_1},{Z_2},{Z_3}} \right) \in {\mathbb{R}^3}: {\left( {{Z_1} - \tau _{i1}} \right)^2} + {\left( {{Z_2} - \tau _{i2}} \right)^2} + {\left( {{Z_3} - \tau _{i3}} \right)^2} \leqslant {r{\tau_i}^2}
\end{gathered}  \right\},
\end{equation*}
for $i=1,\ldots,n$.
\end{definition}

To ensure the attraction of each ${\mathcal{P}{v_i}}$ to its designated target, the target attraction function is:
\begin{equation}\label{Attraction_for_carlike_robot_evric}
V{v_i} = \frac{1}{2}\left[ {{{\left( {x{v_i} - {\tau _{i1}}} \right)}^2} + {{\left( {y{v_i} - {\tau _{i2}}} \right)}^2} + {{\left( {z{v_i} - {\tau _{i3}}} \right)}^2} + \nu {v_i}^2 + \omega {v_i}^2 + \mu {v_i}^2} \right],
\end{equation}
for $i=1,\ldots,n$.
%
\subsection{Auxiliary Functions}
\subsubsection{Auxiliary function for $\mathcal{P}_i$}
We need to guarantee that each ${\mathcal{P}_i}$ converges to its designated mobile target and also ensure that the non- linear controllers vanish at the target such that the energy of the system is zero. Hence, we define the auxiliary function as:
\begin{equation}\label{Aux_for_UAV_pm_evric}
 G{p_i} = \frac{1}{2}\left[ {{{\left( {x{p_i} - x{v_i}} \right)}^2} + {{\left( {y{p_i} - y{v_i}} \right)}^2} + {{\left( {z{p_i} - \left( {z{v_i} + r{v_i} + r{p_i}} \right)} \right)}^2}} \right],
\end{equation}
for $i=1,\ldots,n$.
%
\subsubsection{Auxiliary function for $\mathcal{V}_i$}
We need to guarantee that each ${\mathcal{V}_i}$ converges to its designated fixed target and also ensure that the non- linear controllers vanish at the target such that the energy of the system is zero. Hence, we define the auxiliary function as:
\begin{equation}\label{Aux_for_carlike_robot_evric}
G{v_i} = \frac{1}{2}\left[ {{{\left( {x{v_i} - {\tau _{i1}}} \right)}^2} + {{\left( {y{v_i} - {\tau _{i2}}} \right)}^2} + {{\left( {z{v_i} - {\tau _{i3}}} \right)}^2}} \right],
\end{equation}
for $i=1,\ldots,n$.
%
\subsection{Obstacle Avoidance Functions}
While en route to its target, every mobile robot will need to avoid certain obstacles, either fixed or moving. In this section, functions are constructed which acts as obstacle avoidance functions for $\mathcal{P}_i$ and $\mathcal{V}_i$.
\subsubsection{Workspace boundary limitations}
A $3$-dimensional workspace is considered in which the robots will be confined to at all time $t \geqslant 0$. Thus the following is defined:
\begin{definition}
The workspace, WS, is a fixed, bounded and closed cuboidal region.
\begin{equation*}
WS = \left\{ {\left( {{Z_1},{Z_2},{Z_3}} \right) \in {\mathbb{R}^3}:0 \leqslant {Z_1} \leqslant {\eta _1},\,0 \leqslant {Z_2} \leqslant {\eta _2},\,0 \leqslant {Z_3} \leqslant {\eta _3}} \right\},
\end{equation*}
for some $\left\{ {{\eta _1} > 0,{\eta _2} > 0,{\eta _3} > 0} \right\}$.
\end{definition}
These rectangular boundary walls act as fixed obstacles which the mobile robots need to avoid at all times. Hence we construct the positive obstacle avoidance functions.
\paragraph{Boundary avoidance by $\mathcal{P}_i$}
The following obstacle avoidance functions is designed for each $\mathcal{P}_i$ to avoid the fixed boundary walls:
\begin{eqnarray}
  \left. \begin{gathered}
  W{p_{i1}} = x{p_i} - r{p_i}, W{p_{i4}} = {\eta_1} - \left( {x{p_i} + r{p_i}} \right),\\
  W{p_{i2}} = y{p_i} - r{p_i}, W{p_{i5}} = {\eta_2} - \left( {y{p_i} + r{p_i}} \right),\\
  W{p_{i3}} = z{p_i} - r{p_i}, W{p_{i6}} = {\eta_3} - \left( {z{p_i} + r{p_i}} \right),
\end{gathered}  \right\}
\end{eqnarray}
for $i=1,\ldots,n$.
%
\paragraph{Boundary avoidance by $\mathcal{V}_i$}
The following obstacle avoidance functions is designed for each $\mathcal{V}_i$ to avoid the fixed boundary walls:
\begin{eqnarray}
 \left. \begin{gathered}
  W{v_{i1}} = x{v_i} - r{v_i},\,\,W{v_{i4}} = {\eta_1} - \left( {x{v_i} + r{v_i}} \right),\\
  W{v_{i2}} = y{v_i} - r{v_i},\,\,W{v_{i5}} = {\eta_2} - \left( {y{v_i} + r{v_i}} \right),\\
  W{v_{i3}} = z{v_i} - r{v_i},\,\,W{v_{i6}} = {\eta_3} - \left( {z{v_i} + r{v_i}} \right),
\end{gathered}  \right\}
\end{eqnarray}
for $i=1,\ldots,n$.
%
\subsubsection{Cylindrical obstacles}
Cylindrical obstacles are used to exemplify buildings in a \emph{smart city} which the robots need to avoid while in motion towards their target. The $\mathcal{P}_i$ and $\mathcal{V}_i$ needs to avoid the surface wall of the cylinders. For each robot to avoid these curved walls, we use the MDT, which calculates the minimum distance between the robot and the surface wall of the $k$th cylinder. The following definition is proposed:
\begin{definition}
  The surface wall of the $k$th cylinder in the ${Z_1}{Z_2}{Z_3}$ plane is between the coordinates $\left( {{a_k},{b_k},{c_{k1}}} \right)$ and $\left( {{a_k},{b_k},{c_{k2}}} \right)$, with center $\left( {{a_k},{b_k},{c_{k1}}} \right)$, radius $rc_k$ and height of $\left( {{c_{k2}} - {c_{k1}}} \right)$ for $k=1,\ldots,m$ where $m$ is the number of cylindrical obstacles. Hence the parametric representation of the surface wall of the $k$th cylinder is given as $C{x_k} = {a_k} \pm r{c_k}.\cos {{\theta _k}} , C{y_k} = {b_k} \pm r{c_k}.\sin  {{\theta _k}} $ and $C{z_k} = {c_{k1}} + {\lambda _k}\left( {{c_{k2}} - {c_{k1}}} \right)$ where ${\theta _k} \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)$ and ${\lambda _k} \in \left[ {0,1} \right]$.
\end{definition}

\paragraph{Cylindrical obstacle avoidance by $\mathcal{P}_i$}
In order for each $\mathcal{P}_i$ to avoid the surface wall of the $k$th cylindrical obstacle, the MDT is utilized for finding the minimum distance between the center of each $\mathcal{P}_i$ and the surface wall of the $k$th cylinder. Considering the point $\left( Cx{p_{ik}}, Cy{p_{ik}}, Cz{p_{ik}} \right)$ to be any point on the surface wall of the $k$th cylinder that each $\mathcal{P}_i$ needs to avoid, we get the following parametric representation of the point:
\begin{equation*}
Cx{p_{ik}} = {a_k} \pm r{c_k}.\cos \theta {p_{ik}},Cy{p_{ik}} = {b_k} \pm r{c_k}.\sin \theta {p_{ik}},Cz{p_{ik}} = {c_{k1}} + \lambda {p_{ik}}\left( {{c_{k2}} - {c_{k1}}} \right)
\end{equation*}
where
\begin{equation*}
\theta {p_{ik}} = {\tan ^{ - 1}}\left( {\frac{{y{p_i} - {b_k}}}{{x{p_i} - {a_k}}}} \right) \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)
\end{equation*}
and
\begin{equation*}
\lambda {p_{ik}} = \frac{1}{{\left( {{c_{k2}} - {c_{k1}}} \right)}}\left( {z{p_i} - {c_{k1}}} \right)\in \left[ {0,1} \right],
\end{equation*}
whose saturation function is
\begin{equation*}
{\lambda {p_{ik}}} = \left\{ \begin{gathered}
  0\,\,\,\,\,,\,$if$\,\,{\lambda {p_{ik}}} < 0 \hfill \\
  {\lambda {p_{ik}}},\,$if$\,\,0 \leqslant {\lambda {p_{ik}}} \leqslant 1 \hfill \\
  1\,\,\,\,\,,\,$if$\,\,{\lambda {p_{ik}}} > 1\, \hfill \\
\end{gathered}  \right..
\end{equation*}
In order to avoid the whole cylinder, each $\mathcal{P}_i$ needs to avoid these closest points on the surface wall of the $k$th cylinder for all time $t \geqslant 0$. Thus we construct the following obstacle avoidance function:
\begin{equation}\label{cylinder_avoidance_UAV_robot_evric}
CO{p_{ik}} = \frac{1}{2}\left( {{{\left( {x{p_i} - Cx{p_{ik}}} \right)}^2} + {{\left( {y{p_i} - Cy{p_{ik}}} \right)}^2} + {{\left( {z{p_i} - Cz{p_{ik}}} \right)}^2} - r{p_i}^2} \right),
\end{equation}
for $k = 1,\ldots,m$ and $i = 1,\ldots,n$.
%
\paragraph{Cylindrical obstacle avoidance by $\mathcal{V}_i$}
Every $i$th car-like robot needs to avoid the surface wall of the $k$th cylindrical obstacle. Thus the MDT is adopted to find the minimum distance between the center of each $\mathcal{V}_i$ and the surface wall of the $k$th cylinder. Letting $\left( Cx{v_{ik}}, Cy{v_{ik}}, Cz{v_{ik}} \right)$ to be any point on the surface wall of the $k$th cylinder that each $\mathcal{V}_i$ needs to avoid, the following parametric representation of the point is derived:
\begin{equation*}
Cx{v_{ik}} = {a_k} \pm r{c_k}.\cos \theta {v_{ik}},Cy{v_{ik}} = {b_k} \pm r{c_k}.\sin \theta {v_{ik}},Cz{v_{ik}} = {c_{k1}} + \lambda {v_{ik}}\left( {{c_{k2}} - {c_{k1}}} \right)
\end{equation*}
where
\begin{equation*}
\theta {v_{ik}} = {\tan ^{ - 1}}\left( {\frac{{y{v_i} - {b_k}}}{{x{v_i} - {a_k}}}} \right) \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)
\end{equation*}
and
\begin{equation*}
\lambda {v_{ik}} = \frac{1}{{\left( {{c_{k2}} - {c_{k1}}} \right)}}\left( {z{v_i} - {c_{k1}}} \right)\in \left[ {0,1} \right],
\end{equation*}
whose saturation function is
\begin{equation*}
{\lambda {v_{ik}}} = \left\{ \begin{gathered}
  0\,\,\,\,\,\,\,\,,\,$if$\,\,{\lambda {v_{ik}}} < 0 \hfill \\
  {\lambda {v_{ik}}},\,$if$\,\,0 \leqslant {\lambda {v_{ik}}} \leqslant 1 \hfill \\
  1\,\,\,\,\,\,\,\,,\,$if$\,\,{\lambda {v_{ik}}} > 1\, \hfill \\
\end{gathered}  \right..
\end{equation*}
In order to avoid the whole cylinder, each $\mathcal{V}_i$ needs to avoid these closest points on the surface wall of the $k$th cylinder for all time $t \geqslant 0$. Thus we construct the following obstacle avoidance function:
\begin{equation}\label{cylinder_avoidance_UAV_robot_evric}
CO{v_{ik}} = \frac{1}{2}\left[ {{{\left( {x{v_i} - Cx{v_{ik}}} \right)}^2} + {{\left( {y{v_i} - Cy{v_{ik}}} \right)}^2} + {{\left( {z{v_i} - Cz{v_{ik}}} \right)}^2} - r{v_i}^2} \right],
\end{equation}
for $k = 1,\ldots,m$ and $i = 1,\ldots,n$.
%
\subsubsection{Moving Obstacles}
Each UAV point-mass becomes a moving obstacle for every other UAV point-masses and each car-like mobile platform acts as a moving obstacle for every other car-like robots in the bounded workspace. Therefore we design the obstacle avoidance functions to ensure the moving robots avoid each other.
\paragraph{Moving obstacle avoidance by $\mathcal{P}_i$}
For each $\mathcal{P}_i$ to avoid every $\mathcal{P}_j$, the following obstacle avoidance function is designed:
\begin{equation}\label{Inter_collision_Pp_evric}
M{p_{ij}}\left( {\mathbf{x}} \right) = \frac{1}
{2}\left[ \begin{gathered}
  {\left( {x{p_i} - x{p_j}} \right)^2} + {\left( {y{p_i} - y{p_j}} \right)^2} + {\left( {z{p_i} - z{p_j}} \right)^2}  - {\left( {2\,r{p_i}} \right)^2}
\end{gathered}  \right],
\end{equation}
where $i,j=1,\ldots,n$ and ${j\neq i}$. This is a $2$-way avoidance scheme.
%
\paragraph{Moving obstacle avoidance by $\mathcal{V}_i$}
For each $\mathcal{V}_i$ to avoid every $\mathcal{V}_j$, the following obstacle avoidance function is designed:
\begin{equation}\label{Inter_collision_Pv_evric}
M{v_{ij}}\left( {\mathbf{x}} \right) = \frac{1}{2}\left[ \begin{gathered}
  {\left( {x{v_i} - x{v_j}} \right)^2} + {\left( {y{v_i} - y{v_j}} \right)^2} + {\left( {z{v_i} - z{v_j}} \right)^2} - {\left( {2\,r{v_i}} \right)^2}
\end{gathered}  \right],
\end{equation}
where $i,j=1,\ldots,n$ and ${j\neq i}$. This is also a $2$-way avoidance scheme.

\subsubsection{Modulus bound on velocities of the car-like mobile landing platforms}
For security and inherent limitations, the translational speed and  the steering angle of the car-like robot are restricted. We consider ${\nu _{\max }} > 0,\,{\omega _{\max }} > 0$ and ${\mu _{\max }} > 0$ to be the maximum speeds in $Z_1, Z_2, Z_3$ directions, respectively, and ${\phi _{\max }}$ to be the maximum steering angle where $0 < {\phi _{\max }} < \frac{\pi }{2}$. As shown in \cite{sharma2007carlike, jairaj2018}, we impose the following additional constraints on the translational velocity, $\nu {v_i}$, the rotational velocity, $\omega {v_i}$, and the rate of the change of altitude, $\mu {v_i}$ of the car-like robot:
%
\begin{equation}\label{modulus_bound_constraint_evric}
\left| {\nu {v_i}} \right| < {\nu _{\max }},\left| {\omega {v_i}} \right| < {\omega _{\max }},\left| {\mu {v_i}} \right| < {\mu _{\max }}.
\end{equation}
%
To factor these dynamic constraints into the control schemes, \emph{artificial obstacles} need to be constructed and avoided to ensure the desired result is achieved \cite{pappas1995}. The following artificial obstacles are constructed to control each car-like robot within these constraints:

\[\begin{gathered}
  A{O_{i1}} = \left\{ {\nu {v_i} \in \mathbb{R}:\left| {\nu {v_i}} \right| \geqslant  {\nu _{\max }}} \right\}, \hfill \\
  A{O_{i2}} = \left\{ {\omega {v_i} \in \mathbb{R}:\left| {\omega {v_i}} \right| \geqslant  {\omega _{\max }}} \right\}, \hfill \\
  A{O_{i3}} = \left\{ {\mu {v_i} \in \mathbb{R}:\left| {\mu {v_i}} \right| \geqslant  {\mu _{\max }}} \right\}. \hfill \\
\end{gathered} \]

Therefore, to ensure the car-like robot adheres to the constraint (\ref{modulus_bound_constraint_evric}) and avoids the artificial obstacles, the following avoidance function is designed:
\begin{equation}\label{modulus_bound_function_evric}
\left. \begin{gathered}
  {U_{i1}} = \frac{1}{2}\left( {{\nu _{\max }} - \nu {v_i}} \right)\left( {{\nu _{\max }} + \nu {v_i}} \right) \hfill \\
  {U_{i2}} = \frac{1}{2}\left( {{\omega _{\max }} - \omega {v_i}} \right)\left( {{\omega _{\max }} + \omega {v_i}} \right) \hfill \\
  {U_{i3}} = \frac{1}{2}\left( {{\mu _{\max }} - \mu {v_i}} \right)\left( {{\mu _{\max }} + \mu {v_i}} \right)
\end{gathered}  \right\}
\end{equation}
for $i=1,\ldots,n$.

\section{Nonlinear Controller Design}\label{NLCD_evric}
In accordance with LbCS, we design the nonlinear control laws that governs system (\ref{first_order_ODE_evric}). We begin by constructing the Lyapunov function and then extracting the control laws using the time-derivative of the Lyapunov function that will govern the motion of the dynamic robotic system in the constrained environment.

\subsection{Lyapunov Function}
We design the Lyapunov function for system (\ref{first_order_ODE_evric}) by combining the target attraction functions and the obstacle avoidance functions. When these obstacle avoidance functions appear in the denominator of appropriate terms in the Lyapunov function, they form the repulsive potential field functions, in the presence of some \emph{tuning parameters}. Thus, the following \emph{tuning parameters} is introduced:
\begin{itemize}
  \item [{(i)}] $\alpha {p_{ik}} > 0, k = 1,...,m$, for $\mathcal{P}_i$ to avoid the surface wall of the $k$th cylindrical obstacle;
  \item [{(ii)}] $\alpha {v_{ik}} > 0, k = 1,...,m$, for $\mathcal{V}_i$ to avoid the surface wall of the $k$th cylindrical obstacle;
  \item [{(iii)}] $\beta _{is} > 0, s = 1,...,6$, for $\mathcal{P}_i$ to avoid the $s$th boundary wall of the workspace;
  \item [{(iv)}] $\gamma _{is} > 0, s = 1,...,6$, for $\mathcal{V}_i$ to avoid the $s$th boundary wall of the workspace;
  \item [{(v)}] $\zeta _{ij} > 0$, $j=1,\ldots,n$, $ j \ne i$, for $\mathcal{P}_i$ to avoid the $j$th mobile UAV robot;
  \item [{(vi)}] $\psi _{ij} > 0$, $j=1,\ldots,n$, $ j \ne i$, for $\mathcal{V}_i$ to avoid the $j$th mobile landing platform,
  \item [{(vii)}] $\varphi _{ih} > 0$, $h=1,\ldots,3$, for $\mathcal{V}_i$ to avoid the $h$th artificial obstacle,
\end{itemize}
for $i= 1,\ldots,n$. The tentative Lyapunov function for system (\ref{first_order_ODE_evric}) is defined as:
%
\small
\begin{equation}\label{Lyapunov_evric}
L\left( {\mathbf{x}} \right) = \displaystyle\sum\limits_{i = 1}^n {\left[ \begin{gathered}
V{p_i}\left( {\mathbf{x}} \right) + G{p_i}\left( {\mathbf{x}} \right)\left( {\displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {p_{ik}}}}
{{CO{p_{ik}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{s = 1}^6 {\frac{{{\beta _{is}}}}{{W{p_{is}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\zeta _{ij}}}}
{{M{p_{ij}}\left( {\mathbf{x}} \right)}}} } } \right) +   \\
V{v_i}\left( {\mathbf{x}} \right) + G{v_i}\left( {\mathbf{x}} \right)\left( {\displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {v_{ik}}}}
{{CO{v_{ik}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{s = 1}^6 {\frac{{{\gamma _{is}}}}{{W{v_{is}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\psi _{ij}}}}
{{M{v_{ij}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{h = 1}^3 {\frac{{{\varphi _{ih}}}}
{{{U_{ih}}\left( {\mathbf{x}} \right)}}} } } \right)  \\
\end{gathered}  \right]}
\end{equation}
\normalsize
which is positive over the domain $\mathbb{D}\left( L (\mathbf{x})\right) = \{ \mathbf{x} \in {\mathbb{R}^{3n}}:W{p_{is}}\left( \mathbf{x} \right) > 0,\,s = 1,\ldots,6; W{v_{is}}\left( \mathbf{x} \right) > 0,\,s = 1,\ldots,6; CO{p_{ik}}(\mathbf{x}) > 0, k = 1,\ldots,m; CO{v_{ik}}(\mathbf{x}) > 0, k = 1,\ldots,m;\,\,M{p_{ij}}(\mathbf{x}) > 0, j \ne i; M{v_{ij}}(\mathbf{x}) > 0, j \ne i\; U_{ih}(\mathbf{x}) > 0, h = 1,\ldots,3 \}$ for $i = 1,\ldots,n $.
%
\subsection{Nonlinear Controller Design}
We extract the acceleration control laws of the kinodynamic system by differentiating the components of $L\left( {\mathbf{x}} \right)$ in equation (\ref{Lyapunov_evric}) with respect to time. The components of the control inputs are as follows:
\small
\begin{equation*}
\begin{gathered}
  {f_{i1}} = \left( {1 + \displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {p_{ik}}}}
{{CO{p_{ik}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{s = 1}^6 {\frac{{{\beta _{is}}}}
{{W{P_{is}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\zeta _{ij}}}}
{{M{p_{ij}}\left( {\mathbf{x}} \right)}}} } } \right)\left( {x{p_i} - x{v_i}} \right) \hfill \\
  \,\,\,\,\,\,\,\,\,\,\,\, - G{p_i}\left( {\mathbf{x}} \right)\left( {\frac{{{\beta _{i1}}}}
{{{{\left( {W{P_{i1}}\left( {\mathbf{x}} \right)} \right)}^2}}}\, - \frac{{{\beta _{i4}}}}
{{{{\left( {W{P_{i4}}\left( {\mathbf{x}} \right)} \right)}^2}}} + 2\displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\zeta _{ij}}}}
{{{{\left( {M{p_{ij}}\left( {\mathbf{x}} \right)} \right)}^2}}}\left( {x{p_i} - x{p_j}} \right)} } \right) \hfill \\
  \,\,\,\,\,\,\,\,\,\,\,\, - G{p_i}\left( {\mathbf{x}} \right)\displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {p_{ik}}}}
{{{{\left( {CO{p_{ik}}\left( {\mathbf{x}} \right)} \right)}^2}}}\left[ \begin{gathered}
  \left( {x{p_i} - Cx{p_{ik}}} \right)\left( {1 \mp r{c_k}.\sin  {\theta {p_k}} \frac{{y{p_i} - {b_k}}}
{{{{\left( {x{p_i} - {a_k}} \right)}^2} + {{\left( {y{p_i} - {b_k}} \right)}^2}}}} \right) \hfill \\
   + \left( {y{p_i} - Cy{p_{ik}}} \right)\left( { \pm r{c_k}.\cos  {\theta {p_k}} \frac{{y{p_i} - {b_k}}}
{{{{\left( {x{p_i} - {a_k}} \right)}^2} + {{\left( {y{p_i} - {b_k}} \right)}^2}}}\,} \right) \hfill \\
\end{gathered}  \right]}  \hfill \\
\end{gathered}
\end{equation*}
%
\begin{equation*}
\begin{gathered}
  {f_{i2}} = \left( {1 + \displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {p_{ik}}}}
{{CO{p_{ik}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{s = 1}^6 {\frac{{{\beta _{is}}}}
{{W{P_{is}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\zeta _{ij}}}}
{{M{p_{ij}}\left( {\mathbf{x}} \right)}}} } } \right)\left( {y{p_i} - y{v_i}} \right) \hfill \\
  \,\,\,\,\,\,\,\,\,\,\, - G{p_i}\left( {\mathbf{x}} \right)\left( {\frac{{{\beta _{i2}}}}
{{{{\left( {W{P_{i2}}\left( {\mathbf{x}} \right)} \right)}^2}}}\, - \frac{{{\beta _{i5}}}}
{{{{\left( {W{P_{i5}}\left( {\mathbf{x}} \right)} \right)}^2}}} + 2\displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\zeta _{ij}}}}
{{{{\left( {M{p_{ij}}\left( {\mathbf{x}} \right)} \right)}^2}}}\left( {y{p_i} - y{p_j}} \right)} } \right) \hfill \\
\,\,\,\,\,\,\,\,\,\,\, - G{p_i}\left( {\mathbf{x}} \right)\displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {p_{ik}}}}
{{{{\left( {CO{p_{ik}}\left( {\mathbf{x}} \right)} \right)}^2}}}\left[ \begin{gathered}
  \left( {x{p_i} - Cx{p_{ik}}} \right)\left( { \pm r{c_k}.\sin  {\theta {p_k}} \frac{{x{p_i} - {a_k}}}
{{{{\left( {x{p_i} - {a_k}} \right)}^2} + {{\left( {y{p_i} - {b_k}} \right)}^2}}}} \right) \hfill \\
   + \left( {y{p_i} - Cy{p_{ik}}} \right)\left( {1 \mp r{c_k}.\cos  {\theta {p_k}} \frac{{x{p_i} - {a_k}}}
{{{{\left( {x{p_i} - {a_k}} \right)}^2} + {{\left( {y{p_i} - {b_k}} \right)}^2}}}\,} \right) \hfill \\
\end{gathered}  \right]}  \hfill \\
\end{gathered}
\end{equation*}
  %
  \begin{equation*}
\begin{gathered}
  {f_{i3}} = \left( {1 + \displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {p_{ik}}}}
{{CO{p_{ik}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{s = 1}^6 {\frac{{{\beta _{is}}}}
{{W{P_{is}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\zeta _{ij}}}}
{{M{p_{ij}}\left( {\mathbf{x}} \right)}}} } } \right)\left( {z{p_i} - z{v_i}} \right) \hfill \\
  \,\,\,\,\,\,\,\,\,\,\,\, - G{p_i}\left( {\mathbf{x}} \right)\left( {\frac{{{\beta _{i3}}}}
{{{{\left( {W{P_{i3}}\left( {\mathbf{x}} \right)} \right)}^2}}}\, - \frac{{{\beta _{i6}}}}
{{{{\left( {W{P_{i6}}\left( {\mathbf{x}} \right)} \right)}^2}}} + 2\displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\zeta _{ij}}}}
{{{{\left( {M{p_{ij}}\left( {\mathbf{x}} \right)} \right)}^2}}}\left( {z{p_i} - z{p_j}} \right)} } \right) \hfill \\
\end{gathered}
\end{equation*}
%
\begin{equation*}
\begin{gathered}
  {f_{i4}} = \left( {1 + \displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {v_{ik}}}}
{{CO{v_{ik}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{s = 1}^6 {\frac{{{\gamma _{is}}}}
{{W{V_{is}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum\limits_{h = 1}^3 {\frac{{{\varphi _{ih}}}}
{{{U_{ih}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\psi _{ij}}}}
{{M{v_{ij}}\left( {\mathbf{x}} \right)}}} } } } \right)\left( {x{v_i} - {\tau _{i1}}} \right) \hfill \\
  \,\,\,\,\,\,\,\,\,\,\,\,\, - \left( {1 + \displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {p_{ik}}}}
{{CO{p_{ik}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{s = 1}^6 {\frac{{{\beta _{is}}}}
{{W{P_{is}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\zeta _{ij}}}}
{{M{p_{ij}}\left( {\mathbf{x}} \right)}}} } } \right)\left( {x{p_i} - x{v_i}} \right) \hfill \\
  \,\,\,\,\,\,\,\,\,\,\,\,\, - G{v_i}\left( {\mathbf{x}} \right)\left( {\frac{{{\gamma _{i1}}}}
{{{{\left( {W{V_{i1}}\left( {\mathbf{x}} \right)} \right)}^2}}} - \frac{{{\gamma _{i4}}}}
{{{{\left( {W{V_{i4}}\left( {\mathbf{x}} \right)} \right)}^2}}} + 2\displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\psi _{ij}}}}
{{{{\left( {M{v_{ij}}\left( {\mathbf{x}} \right)} \right)}^2}}}\left( {x{v_i} - x{v_j}} \right)} } \right) \hfill \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\, - G{v_i}\left( {\mathbf{x}} \right)\displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {v_{ik}}}}
{{{{\left( {CO{v_{ik}}\left( {\mathbf{x}} \right)} \right)}^2}}}\left[ \begin{gathered}
  \left( {x{v_i} - Cx{v_{ik}}} \right)\left( {1 \mp r{c_k}.\sin  {\theta {v_k}} \frac{{y{v_i} - {b_k}}}
{{{{\left( {x{v_i} - {a_k}} \right)}^2} + {{\left( {y{v_i} - {b_k}} \right)}^2}}}} \right) \hfill \\
   + \left( {y{v_i} - Cy{v_{ik}}} \right)\left( { \pm r{c_k}.\cos  {\theta {v_k}} \frac{{y{v_i} - {b_k}}}
{{{{\left( {x{v_i} - {a_k}} \right)}^2} + {{\left( {y{v_i} - {b_k}} \right)}^2}}}\,} \right) \hfill \\
\end{gathered}  \right]}  \hfill \\
\end{gathered}
\end{equation*}
%
\begin{equation*}
\begin{gathered}
{f_{i5}} = \left( {1 + \displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {v_{ik}}}}
{{CO{v_{ik}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{s = 1}^6 {\frac{{{\gamma _{is}}}}
{{W{V_{is}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum\limits_{h = 1}^3 {\frac{{{\varphi _{ih}}}}
{{{U_{ih}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\psi _{ij}}}}
{{M{v_{ij}}\left( {\mathbf{x}} \right)}}} } } } \right)\left( {y{v_i} - {\tau _{i2}}} \right) \hfill \\
\,\,\,\,\,\,\,\,\,\,\,\,- \left( {1 + \displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {p_{ik}}}}
{{CO{p_{ik}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{s = 1}^6 {\frac{{{\beta _{is}}}}
{{W{P_{is}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\zeta _{ij}}}}
{{M{p_{ij}}\left( {\mathbf{x}} \right)}}} } } \right)\left( {y{p_i} - y{v_i}} \right) \hfill \\
  \,\,\,\,\,\,\,\,\,\,\,\, - G{v_i}\left( {\mathbf{x}} \right)\left( {\frac{{{\gamma _{i2}}}}
{{{{\left( {W{V_{i2}}\left( {\mathbf{x}} \right)} \right)}^2}}} - \frac{{{\gamma _{i5}}}}
{{{{\left( {W{V_{i5}}\left( {\mathbf{x}} \right)} \right)}^2}}} + 2\displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\psi _{ij}}}}
{{{{\left( {M{v_{ij}}\left( {\mathbf{x}} \right)} \right)}^2}}}\left( {y{v_i} - y{v_j}} \right)} } \right) \hfill \\
  \,\,\,\,\,\,\,\,\,\,\,\, - G{v_i}\left( {\mathbf{x}} \right)\displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {v_{ik}}}}
{{{{\left( {CO{v_{ik}}\left( {\mathbf{x}} \right)} \right)}^2}}}\left[ \begin{gathered}
  \left( {x{v_i} - Cx{v_{ik}}} \right)\left( { \pm r{c_k}.\sin  {\theta {v_k}} \frac{{x{v_i} - {a_k}}}
{{{{\left( {x{v_i} - {a_k}} \right)}^2} + {{\left( {y{v_i} - {b_k}} \right)}^2}}}} \right) \hfill \\
   + \left( {y{v_i} - Cy{v_{ik}}} \right)\left( {1 \mp r{c_k}.\cos  {\theta {v_k}} \frac{{x{v_i} - {a_k}}}
{{{{\left( {x{v_i} - {a_k}} \right)}^2} + {{\left( {y{v_i} - {b_k}} \right)}^2}}}\,} \right) \hfill \\
\end{gathered}  \right]}  \hfill \\
\end{gathered}
\end{equation*}
  %
\begin{equation*}
\begin{gathered}
{f_{i6}} = \left( {1 + \displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {v_{ik}}}}
{{CO{v_{ik}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{s = 1}^6 {\frac{{{\gamma _{is}}}}
{{W{V_{is}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum\limits_{h = 1}^3 {\frac{{{\varphi _{ih}}}}
{{{U_{ih}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\psi _{ij}}}}
{{M{v_{ij}}\left( {\mathbf{x}} \right)}}} } } } \right)\left( {z{v_i} - {\tau _{i3}}} \right) \hfill \\
  \,\,\,\,\,\,\,\,\,\,\, - \left( {1 + \displaystyle\sum\limits_{k = 1}^m {\frac{{\alpha {p_{ik}}}}
{{CO{p_{ik}}\left( {\mathbf{x}} \right)}}}  + \displaystyle\sum\limits_{s = 1}^6 {\frac{{{\beta _{is}}}}
{{W{P_{is}}\left( {\mathbf{x}} \right)}} + \displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\zeta _{ij}}}}
{{M{p_{ij}}\left( {\mathbf{x}} \right)}}} } } \right)\left( {z{p_i} - z{v_i}} \right) \hfill \\
  \,\,\,\,\,\,\,\,\,\,\, - G{v_i}\left( {\mathbf{x}} \right)\left( {\frac{{{\gamma _{i3}}}}
{{{{\left( {W{V_{i3}}\left( {\mathbf{x}} \right)} \right)}^2}}} - \frac{{{\gamma _{i6}}}}
{{{{\left( {W{V_{i6}}\left( {\mathbf{x}} \right)} \right)}^2}}} + 2\displaystyle\sum_{j=1\atop i\neq j}^{n} {\frac{{{\psi _{ij}}}}
{{{{\left( {M{v_{ij}}\left( {\mathbf{x}} \right)} \right)}^2}}}\left( {z{v_i} - z{v_j}} \right)} } \right) \hfill \\
\end{gathered}
\end{equation*}
%
\begin{equation*}
{f_{i7}} = 1 + G{v_i}\left( x \right)\frac{{{\varphi _{i1}}}}{{{{({U_{i1}})}^2}}}
\end{equation*}
%
\begin{equation*}
{f_{i8}} = 1 + G{v_i}\left( x \right)\frac{{{\varphi _{i2}}}}{{{{({U_{i2}})}^2}}}
\end{equation*}
%
\begin{equation*}
{f_{i9}} = 1 + G{v_i}\left( x \right)\frac{{{\varphi _{i3}}}}{{{{({U_{i3}})}^2}}}
\end{equation*}
%
By introducing the convergence parameters to be ${\delta _{ib}} > 0,\,$ for $b = 1,\ldots,6$, we get the acceleration controllers of the form:
\begin{equation}\label{acceleration_controllers_evric}
\left. \begin{gathered}
  {\sigma _{i1}} =  - \left( {{\delta _{i1}}\,\nu {p_i}\, + {f_{i1}}} \right) \hfill \\
  {\sigma _{i2}} =  - \left( {{\delta _{i2}}\,\omega {p_i}\, + {f_{i2}}} \right) \hfill \\
  {\sigma _{i3}} =  - \left( {{\delta _{i3}}\,\mu {p_i}\, + {f_{i3}}} \right) \hfill \\
  {\sigma _{i4}} =  - \frac{{\left( {{\delta _{i4}}\,\nu {v_i}\, + {f_{i4}}\cos  {\phi {v_i}}  + {f_{i5}}\sin  {\phi {v_i}} } \right)}}
{{{f_{i7}}}} \hfill \\
  {\sigma _{i5}} =  - \frac{{\left[ {{\delta _{i5}}\,\omega {v_i}\, + \frac{L}
{2}\left( {{f_{i5}}\cos  {\phi {v_i}}  - {f_{i4}}\sin  {\phi {v_i}} } \right)} \right]}}
{{{f_{i8}}}} \hfill \\
  {\sigma _{i6}} =  - \frac{{\left( {{\delta _{i6}}\,\mu {v_i}\, + {f_{i6}}} \right)}}
{{{f_{i9}}}} \hfill \\
\end{gathered}  \right\}
\end{equation}

\section{Stability Analysis}\label{stability_evric}
This section will illustrate the proof of the system stability of system (\ref{first_order_ODE_evric}).
%
\begin{theorem}
We assume that a fixed point $\mathbf{x}_{e_i}:=\left(\tau_{i1},\tau_{i2},\tau_{i3}\right)\in{\mathbb{R}^{3}}$ is an equilibrium point of the robotic system for $i=1,\ldots,n$. Then $\mathbf{x}_{e}:= (\mathbf{x}_{e_1}, \ldots, \mathbf{x}_{e_n}) \in \mathbb{R}^{3n}$ is a stable equilibrium point of system (\ref{first_order_ODE_evric}).
\end{theorem}
%
\emph{\textbf{Proof.}} Since the Lyapunov function $L\left( {\bf{x}} \right)$ of  system (\ref{first_order_ODE_evric}) is defined, continuous and positive over the
domain $\mathbb{D}\left( L (\mathbf{x})\right) = \{ \mathbf{x} \in {\mathbb{R}^{3n}}:W{p_{is}}\left( \mathbf{x} \right) > 0,\,s = 1,\ldots,6; W{v_{is}}\left( \mathbf{x} \right) > 0,\,s = 1,\ldots,6; CO{p_{ik}}(\mathbf{x}) > 0, k = 1,\ldots,m; CO{v_{ik}}(\mathbf{x}) > 0, k = 1,\ldots,m;\,\,M{p_{ij}}(\mathbf{x}) > 0, j \ne i; M{v_{ij}}(\mathbf{x}) > 0, j \ne i; U_{ih}(\mathbf{x}) > 0, h = 1,\ldots,3 \}$ for $i = 1,\ldots,n$, it can easily be verified that $L\left( {\bf{x}} \right)$ satisfies the following
properties:
%
\begin{enumerate}
    \item  $L\left( {\mathbf{x}} \right)$ is continuous and has first partial derivative in the region $\mathbb{D}$ in the neighborhood of the point $\mathbf{x}_{e}$ of system (\ref{first_order_ODE_evric});
    \item $L(\mathbf{x}_{e})=0$;
    \item  $L(\mathbf{x})>0\;\; \forall{\mathbf{x}}\in{\mathbb{D}(L(\mathbf{x}))}/\mathbf{x}_{e}$.
\end{enumerate}
%--------------------------------------------------------------------------------------------------------------------------------------
Then, along a solution of system (\ref{first_order_ODE_evric}), we have:
%--------------------------------------------------------------------------------------------------------------------------------------
\begin{equation}
  \dot L_{(\ref{first_order_ODE_evric})}\left( {\mathbf{x}} \right) = \displaystyle\sum\limits_{i = 1}^n {\left[ \begin{gathered}
  {f_{i1}}\dot x{p_i} + {f_{i2}}\dot y{p_i} + {f_{i3}}\dot z{p_i} + {f_{i4}}\dot x{v_i} + {f_{i5}}\dot y{v_i} + {f_{i6}}\dot z{v_i} +\hfill \\
  {\sigma _{i1}} \nu {p_i} + {\sigma _{i2}}\omega {p_i}\, + {\sigma _{i3}}\mu {p_i}\, + {f_{i7}}{\sigma _{i4}}\nu {v_i}\, + {f_{i8}} {\sigma _{i5}}\omega {v_i} + {f_{i9}} {\sigma _{i6}}\mu {v_i} \hfill \\
\end{gathered}  \right]}
\end{equation}
%-------------------------------------------------------------------------------------------------------------------------------------
Substituting the controllers given in (\ref{acceleration_controllers_evric}) and the ODE's for system (\ref{first_order_ODE_evric}), we get the following time derivative of $ L\left( {\mathbf{x}} \right)$ which is the semi-negative definite function:
%--------------------------------------------------------------------------------------------------------------------------------------
\begin{equation*}
 \dot L_{(\ref{first_order_ODE_evric})}\left( {\mathbf{x}} \right) =  - \displaystyle\sum\limits_{i = 1}^n {\left[ \begin{gathered}
  {\delta _{i1}}\nu {p_i}^2 + {\delta _{i2}}\omega {p_i}^2 + {\delta _{i3}}\mu {p_i}^2 + {\delta _{i4}}\nu {v_i}^2 + {\delta _{i5}}\omega {v_i}^2 + {\delta _{i6}}\mu {v_i}^2 \hfill \\
\end{gathered}  \right]}  \leqslant 0.
\end{equation*}
%--------------------------------------------------------------------------------------------------------------------------------------
Thus, $\dot{L}_{(\ref{first_order_ODE_evric})}(\mathbf{x})\leq 0$ $\forall \mathbf{x}\in \mathbb{D}(L(\mathbf{x}))$ and $\dot{L}_{(\ref{first_order_ODE_evric})}(\mathbf{x}_e)=0$. Moreover $L(\mathbf{x})\in C^{1}\left(\mathbb{D}(L(\mathbf{x}))\right)$, hence, $L(\mathbf{x})$ is classified as a Lyapunov function for system (\ref{first_order_ODE_evric}) and $\mathbf{x}_e$ is a stable equilibrium point.

\section{Simulation Results}\label{section_sim_evric}
In this section, the simulation results of three scenarios are given where realistic situations are captured to demonstrate the effectiveness and robustness of the nonlinear control laws. In each scenario, UAVs have to land on mobile car-like platforms and navigate together to a final destination, while avoiding fixed and moving obstacles.
\subsection{Scenario 1}
In this scenario, a simple setup is considered using $3$ point-mass robots representing UAVs and $3$ car-like mobile landing platforms in a bounded workspace. The point-mass robots are assigned targets on their respective mobile landing platforms. The point-mass robots navigate towards their mobile landing platforms and converge to their targets. Once landed, the point-mass robots and the car-like mobile landing platforms together navigate towards the mobile landing platforms' predefined targets.In this confined workspace, each $\mathcal{P}_i$ avoids every other $\mathcal{P}_j$ and each $\mathcal{V}_i$ avoids every other $\mathcal{V}_j$ as well for $i=1,\ldots,3$ and $i\neq j$. The $\mathcal{P}_i$ and the $\mathcal{V}_i$ also avoids the boundary walls of the $3$-dimensional workspace. The values of the initial conditions, constraints, and different parameters used are provided in Table \ref{table_1_evric}.

Figure \ref{fig:scenario1_different_motion_views_evric} shows the trajectories at different viewing angles. At times ${\text{t = }}\left\{ {{\text{0, 100, 355, 800, 4000}}} \right\}$, the default, bird's-eye and front views of the trajectories of $\mathcal{P}_i$ and $\mathcal{V}_i$ for $i=1,\ldots,3$ are demonstrated in Figures \ref{fig:scenario1_different_motion_views_evric}(a), \ref{fig:scenario1_different_motion_views_evric}(b) and \ref{fig:scenario1_different_motion_views_evric}(c), respectively.

Figure \ref{fig:scenario1_different_position_views_evric} and \ref{fig:scenario1_different_velocity_views_evric} shows the positions and velocities, respectively, of $3$ point-mass robots and their corresponding mobile landing platforms. It takes $1375$s, $996$s, and $1461$s for $\mathcal{P}_1$, $\mathcal{P}_2$ and $\mathcal{P}_3$, respectively, to converge to their corresponding targets. This has ensured that each $\mathcal{P}_i$ has landed on its corresponding mobile landing platforms. The slight deflections in the $x$ and $y$ directions at around $t=400s$, as shown in Figures \ref{fig:scenario1_different_velocity_views_evric}(a), \ref{fig:scenario1_different_velocity_views_evric}(b), \ref{fig:scenario1_different_velocity_views_evric}(d), \ref{fig:scenario1_different_velocity_views_evric}(e), \ref{fig:scenario1_different_velocity_views_evric}(g), and \ref{fig:scenario1_different_velocity_views_evric}(h), are a result of the inter individual collision avoidance between the point-mass robots and between the mobile landing platforms, respectively.

The behaviour of the corresponding Lyapunov function is shown in Figure \ref{fig:scenario1_lyapunov_and_time_derivative_evric}(a) while Figure \ref{fig:scenario1_lyapunov_and_time_derivative_evric}(b) shows its time derivative along the trajectories of the system.

%%-----------------------------------------------------------------------------
\begin{table}[htbp]
\caption{The parameters used in the numerical simulation for $i =j= 1,\ldots,n$ with $n=3$.}
\label{table_1_evric}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{|p{200pt}|p{195pt}|}
  \hline
Description & Value \\
\hline
\multicolumn{2}{|c|}{\textbf{Initial state of the point-mass mobile robots}} \\
\hline
Workspace &$\eta_{1}=300$, $\eta_{2}=300$, $\eta_{3}=120$\\
%\multicolumn{1}{|c|}{Workspace}                                 &\multicolumn{1}{|c|}{$\eta_{1}=210$, $\eta_{2}=150$, $\eta_{3}=120$}\\
Initial position, radius of $\mathcal{P}_i $ &$(xp_{1},yp_{1},zp_{1})=(30,20,100)$, $rp_{1} = 5$\\
                                                  &$(xp_{2},yp_{2},zp_{2})=(20,150,100)$, $rp_{2} = 5$\\
                                                  &$(xp_{3},yp_{3},zp_{3})=(20,280,100)$, $rp_{3} = 5$\\
Initial position, radius of $\mathcal{V}_i$ &$(xv_{1},yv_{1},zv_{1})=(60,50,3)$, $rv_{1} = 15$\\
                                                  & $(xv_{2},yv_{2},zv_{2})=(30,150,3)$, $rv_{2} = 15$\\
                                                  & $(xv_{3},yv_{3},zv_{3})=(60,260,3)$, $rv_{3} = 15$\\
\hline
 \multicolumn{2}{|c|}{\textbf{Constraints}} \\
 \hline
 Target centre, radius & $(\tau_{11}, \tau_{12}, \tau_{13})=(280,280,3)$, $r \tau_{1}=2$\\
                                  &$(\tau_{21}, \tau_{22}, \tau_{23})=(280,150,3)$, $r \tau_{2}=2$\\
                                  &$(\tau_{31}, \tau_{32}, \tau_{33})=(280,20,3)$, $r \tau_{3}=2$\\
\hline
 \multicolumn{2}{|c|}{\textbf{Control and convergence parameters}} \\
 \hline
  Avoidance of workspace by $\mathcal{P}_i$             & $\beta _{is}= 1$ for $s=1,\ldots, 6$ \\
  Avoidance of workspace by $\mathcal{V}_i$              & $\gamma_{is}= 1$ for $s=1,\ldots, 6$ \\
 Inter individual collision avoidance by $\mathcal{P}_i$       & $\zeta_{ij}= 5$ for $j \ne i$ \\
  Inter individual collision avoidance by $\mathcal{V}_i$       & $\psi_{ij}= 5$ for $j \ne i$ \\
  Modulus bound for $\mathcal{V}_i$                                  &${\varphi _{ih}} = 0.001$ for $h=1,\ldots,3$   \\
  Convergence of $\mathcal{P}_i $ to $Pv_i$                           &$\delta_{i1}=\delta_{i2}=100,\delta_{i3}=1000 $, \\
    Convergence of $\mathcal{V}_i$ to target                &$\delta_{i4}=\delta_{i5}=5000 $, $\delta_{i6}={10^6} $,\\
\hline
\end{tabular}
\end{table}
%%-----------------------------------------------------------------------------


%%-----------------------------------------------------------------------------
\begin{figure*}
$
\centering
\begin{array}{ccc}
{\subfloat[Default 3D motion view.]{\scalebox{0.5}{\includegraphics[width=1.0\textwidth]{scenario1_default_view_evric.pdf}\label{fig_scenario1_default_view_evric}}}}&
{\subfloat[Top 3D motion view.]{\scalebox{0.5}{\includegraphics[width=1.0\textwidth]{scenario1_top_view_evric.pdf}\label{fig_scenario1_top_view_evric}}}}\\
{\subfloat[Front 3D motion view.]{\scalebox{0.5}{\includegraphics[width=1.0\textwidth]{scenario1_front_view_evric.pdf}\label{fig_scenario1_front_view_evric}}}}
\end{array}
$
\caption{Scenario 1. The different viewpoints of the motion of $\mathcal{P}_i$ and $\mathcal{V}_i$, for $i=1,\ldots, 3$, to their respective targets.}
\label{fig:scenario1_different_motion_views_evric}
\end{figure*}
%%-----------------------------------------------------------------------------

%%-----------------------------------------------------------------------------
\begin{figure*}
$
\centering
\begin{array}{ccc}
{\subfloat[The displacement of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the x-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_x11_position_evric.pdf}\label{fig_scenario1_x1_position_evric}}}}&
{\subfloat[The displacement of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the y-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_y11_position_evric.pdf}\label{fig_scenario1_y1_position_evric}}}}&
{\subfloat[The displacement of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the z-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_z11_position_evric.pdf}\label{fig_scenario1_z1_position_evric}}}}\\
{\subfloat[The displacement of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the x-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_x22_position_evric.pdf}\label{fig_scenario1_x2_position_evric}}}}&
{\subfloat[The displacement of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the y-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_y22_position_evric.pdf}\label{fig_scenario1_y2_position_evric}}}}&
{\subfloat[The displacement of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the z-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_z22_position_evric.pdf}\label{fig_scenario1_z2_position_evric}}}}\\
{\subfloat[The displacement of $\mathcal{P}_3$ and $\mathcal{V}_3$ along the x-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_x33_position_evric.pdf}\label{fig_scenario1_x3_position_evric}}}}&
{\subfloat[The displacement of $\mathcal{P}_3$ and $\mathcal{V}_3$ along the y-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_y33_position_evric.pdf}\label{fig_scenario1_y3_position_evric}}}}&
{\subfloat[The displacement of $\mathcal{P}_3$ and $\mathcal{V}_3$ along the z-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_z33_position_evric.pdf}\label{fig_scenario1_z3_position_evric}}}}
\end{array}
$
\caption{Scenario 1. The altitudes and displacements of $\mathcal{P}_i$ and $\mathcal{V}_i$, for $i=1,\ldots, 3$, along the $x-, y-, z-$ axes. The red curves show the respective displacements of the point-mass robots while the blue curves show the respective displacements of the mobile landing platform. The green lines show the positions of the targets in the respective axes.}
\label{fig:scenario1_different_position_views_evric}
\end{figure*}
%%-----------------------------------------------------------------------------


%%-----------------------------------------------------------------------------
\begin{figure*}
$
\centering
\begin{array}{ccc}
{\subfloat[The velocities of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the x-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_velocity_x1_evric.pdf}\label{fig_scenario1_x1_velocity_evric}}}}&
{\subfloat[The velocities of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the y-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_velocity_y1_evric.pdf}\label{fig_scenario1_y1_velocity_evric}}}}&
{\subfloat[The velocities of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the z-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_velocity_z1_evric.pdf}\label{fig_scenario1_z1_velocity_evric}}}}\\
{\subfloat[The velocities of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the x-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_velocity_x2_evric.pdf}\label{fig_scenario1_x2_velocity_evric}}}}&
{\subfloat[The velocities of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the y-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_velocity_y2_evric.pdf}\label{fig_scenario1_y2_velocity_evric}}}}&
{\subfloat[The velocities of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the z-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_velocity_z2_evric.pdf}\label{fig_scenario1_z2_velocity_evric}}}}\\
{\subfloat[The velocities of $\mathcal{P}_3$ and $\mathcal{V}_3$ along the x-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_velocity_x3_evric.pdf}\label{fig_scenario1_x3_velocity_evric}}}}&
{\subfloat[The velocities of $\mathcal{P}_3$ and $\mathcal{V}_3$ along the y-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_velocity_y3_evric.pdf}\label{fig_scenario1_y3_velocity_evric}}}}&
{\subfloat[The velocities of $\mathcal{P}_3$ and $\mathcal{V}_3$ along the z-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario1_velocity_z3_evric.pdf}\label{fig_scenario1_z3_velocity_evric}}}}
\end{array}
$
\caption{Scenario 1. The velocities of $\mathcal{P}_i$ and $\mathcal{V}_i$, for $i=1,\ldots, 3$, along the $x-, y-, z-$ axes. The red curves show the velocities of the point-mass robots while the blue curves show the velocities of the mobile landing platforms.}
\label{fig:scenario1_different_velocity_views_evric}
\end{figure*}
%%-----------------------------------------------------------------------------


%%-----------------------------------------------------------------------------
\begin{figure*}
$
\centering
\begin{array}{cc}
{\subfloat[Lyapunov function.]{\scalebox{0.47}{\includegraphics[width=1.0\textwidth]{scenario1_lyapunov_evric.pdf}\label{scenario1_lyapunov_evric}}}}&
{\subfloat[Lyapunov function time derivative.]{\scalebox{0.47}{\includegraphics[width=1.0\textwidth]{scenario1_lyapunov_derivative_evric.pdf}\label{scenario1_lyapunov_derivative_evric}}}}
\end{array}
$
\caption{Scenario 1. The behaviour of the Lyapunov function and its time derivative.}
\label{fig:scenario1_lyapunov_and_time_derivative_evric}
\end{figure*}
%%-----------------------------------------------------------------------------
\subsection{Scenario 2}
In this scenario, we consider a city-like environment involving $2$ point-mass robots, their respective mobile landing platforms and cylindrical obstacles that mimic buildings. Again, the point-mass robots are assigned targets on their respective mobile landing platforms while the mobile platforms are assigned to their own predefined targets. The point-mass robots while in transit to their defined targets, need to avoid other point-masses and the walls of the cylindrical obstacles. Similarly, the mobile landing platforms avoid other mobile landing platforms and also the walls of the encountering cylindrical obstacles. Once landed, the pair navigate towards the mobile landing platforms predefined targets. The point-mass robots and the mobile landing platforms also need to ensure the avoidance of the closest point on each rectangular wall of the workspace boundary, thus avoiding the entire walls of the boundary. Table \ref{table_2_evric} provides all the values of the initial conditions, constraints and different parameters used in the simulation.

Figure \ref{fig:scenario2_different_motion_views_evric}(a) shows the dafault $3$D view and Figure \ref{fig:scenario2_different_motion_views_evric}(b) shows the top $3$D view while Figure \ref{fig:scenario2_different_motion_views_evric}(c) shows the front $3$D view of the motion of the mobile robots at times ${\text{t = }}\left\{ {{\text{0, 70, 325, 700, 2700}}} \right\}$.

In Figures \ref{fig:scenario2_different_position_views_evric}(a), (b) and Figures \ref{fig:scenario2_different_velocity_views_evric}(a), (b), the disturbance in the displacement and velocity graphs at around ${\text{t = }}\left\{ {{\text{100, 300, 500, 700}}} \right\}$, is a result of the point-mass robot, $\mathcal{P}_1$, and the mobile landing platform, $\mathcal{V}_1$, avoiding the cylindrical obstacles. In Figures \ref{fig:scenario2_different_position_views_evric}(c), (f) and Figures \ref{fig:scenario2_different_velocity_views_evric}(c), (f), at around ${\text{t = }} {{\text{300}}}$ s, the point-mass, $\mathcal{P}_i$ has converged to its predefined target on the mobile landing platform, $\mathcal{V}_i$ for $i=1, 2$.


Figure \ref{fig:scenario2_lyapunov_and_time_derivative_evric}(a) demonstrates the behaviour of the Lyapunov function and its time derivative along the trajectories of the system is shown in Figure \ref{fig:scenario2_lyapunov_and_time_derivative_evric}(b).


%%-----------------------------------------------------------------------------
\begin{table}[htbp]
\caption{The parameters used in the numerical simulation for $i =j= 1,2$ with $m=7$.}
\label{table_2_evric}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{|p{200pt}|p{195pt}|}
  \hline
Description & Value \\
\hline
\multicolumn{2}{|c|}{\textbf{Initial state of the point-mass mobile robots}} \\
\hline
Workspace &$\eta_{1}=300$, $\eta_{2}=300$, $\eta_{3}=120$\\
%\multicolumn{1}{|c|}{Workspace}                                 &\multicolumn{1}{|c|}{$\eta_{1}=210$, $\eta_{2}=150$, $\eta_{3}=120$}\\
Initial position, radius of $\mathcal{P}_i $ &$(xp_{1},yp_{1},zp_{1})=(30,20,100)$, $rp_{1} = 3$\\
                                                  &$(xp_{2},yp_{2},zp_{2})=(20,280,100)$, $rp_{2} = 3$\\
Initial position, radius of $\mathcal{V}_i$ &$(xv_{1},yv_{1},zv_{1})=(60,60,2)$, $rv_{1} = 12$\\
                                                  & $(xv_{2},yv_{2},zv_{2})=(110,230,2)$, $rv_{2} = 12$\\
Cylindrical obstacles - position, radius    &$a_{1} =100, b_{1} = 100, c_{11}=0, c_{12}=100$,\\
                                            & $a_{2} =150, b_{2} = 180, c_{21}=23, c_{22}=45$,\\
                                            & $a_{3} =150, b_{3} = 180, c_{31}=23, c_{32}=45$,\\
                                             & $a_{4} =190, b_{4} = 136, c_{41}=0, c_{42}=23$,\\
                                              & $a_{5} =170, b_{5} = 230, c_{51}=0, c_{52}=23$,\\
                                               & $a_{6} =95, b_{6} = 170, c_{61}=0, c_{62}=23$,\\
                                                & $a_{7} =150, b_{7} = 180, c_{71}=0, c_{72}=3$,\\
                                                 &$rc_{1} =20, rc_{2} = 60, rc_{3} = 55 $\\
                                                  &$rc_{4} = rc_{5} = rc_{6} = rc_{7} = 10$\\
\hline
 \multicolumn{2}{|c|}{\textbf{Constraints}} \\
 \hline
 Target centre, radius & $(\tau_{11}, \tau_{12}, \tau_{13})=(220,270,2)$, $r \tau_{1}=2$\\
                                  &$(\tau_{21}, \tau_{22}, \tau_{23})=(260,30,2)$, $r \tau_{2}=2$\\
\hline
 \multicolumn{2}{|c|}{\textbf{Control and convergence parameters}} \\
 \hline
  Avoidance of workspace by $\mathcal{P}_i$             & $\beta _{is}= 1$ for $s=1,\ldots, 6$ \\
  Avoidance of workspace by $\mathcal{V}_i$              & $\gamma_{is}= 1$ for $s=1,\ldots, 6$ \\
 Avoidance of cylindrical obstacle by $\mathcal{P}_i$       & ${\alpha p _{ik}} = 50$, for $k=1,\ldots,m$   \\
 Avoidance of cylindrical obstacle by $\mathcal{V}_i$       & ${\alpha v _{ik}} = 1$, for $k=1,\ldots,m$   \\
 Inter individual collision avoidance by $\mathcal{P}_i$       & $\zeta_{ij}= 5$ for $j \ne i$ \\
  Inter individual collision avoidance by $\mathcal{V}_i$       & $\psi_{ij}= 10$ for $j \ne i$ \\
  Modulus bound for $\mathcal{V}_i$                                  &${\varphi _{ih}} = 0.001$ for $h=1,\ldots,3$   \\
  Convergence of $\mathcal{P}_i $ to $Pv_i$                           &$\delta_{i1}=5,\delta_{i2}=12, \delta_{13}=60,\delta_{23}=80 $\\
  Convergence of $\mathcal{V}_i$ to target                &$\delta_{i4}=\delta_{i5}=500 $, $\delta_{i6}={10^6} $\\
\hline
\end{tabular}
\end{table}
%%-----------------------------------------------------------------------------
%%-----------------------------------------------------------------------------
\begin{figure*}
$
\centering
\begin{array}{ccc}
{\subfloat[Default 3D motion view.]{\scalebox{0.5}{\includegraphics[width=1.0\textwidth]{scenario2_default_view_evric.pdf}\label{fig_scenario2_default_view_evric}}}}&
{\subfloat[Top 3D motion view.]{\scalebox{0.4}{\includegraphics[width=1.0\textwidth]{scenario2_top_view_evric.pdf}\label{fig_scenario2_top_view_evric}}}}\\
{\subfloat[Front 3D motion view.]{\scalebox{0.5}{\includegraphics[width=1.0\textwidth]{scenario2_front_view_evric.pdf}\label{fig_scenario2_front_view_evric}}}}
\end{array}
$
\caption{Scenario 2. The different viewpoints of $\mathcal{P}_i$ and $\mathcal{V}_i$, for $i=1,2$, in their motion to their targets while avoiding cylindrical obstacles.}
\label{fig:scenario2_different_motion_views_evric}
\end{figure*}
%%-----------------------------------------------------------------------------

%%-----------------------------------------------------------------------------
\begin{figure*}
$
\centering
\begin{array}{ccc}
{\subfloat[The displacement of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the x-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_x1_position_evric.pdf}\label{fig_scenario2_x1_position_evric}}}}&
{\subfloat[The displacement of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the y-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_y1_position_evric.pdf}\label{fig_scenario2_y1_position_evric}}}}&
{\subfloat[The displacement of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the z-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_z1_position_evric.pdf}\label{fig_scenario2_z1_position_evric}}}}\\
{\subfloat[The displacement of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the x-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_x2_position_evric.pdf}\label{fig_scenario2_x2_position_evric}}}}&
{\subfloat[The displacement of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the y-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_y2_position_evric.pdf}\label{fig_scenario2_y2_position_evric}}}}&
{\subfloat[The displacement of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the z-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_z2_position_evric.pdf}\label{fig_scenario2_z2_position_evric}}}}\\
\end{array}
$
\caption{Scenario 2. The altitudes and displacements of $\mathcal{P}_i$ and $\mathcal{V}_i$, for $i=1, 2$, along the $x-, y-, z-$ axes. The red curves show the respective displacements of the point-mass robots while the blue curves show the respective displacements of the mobile landing platforms. The green lines show the positions of the targets in the respective axes.}
\label{fig:scenario2_different_position_views_evric}
\end{figure*}
%%-----------------------------------------------------------------------------


%%-----------------------------------------------------------------------------
\begin{figure*}
$
\centering
\begin{array}{ccc}
{\subfloat[The velocities of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the x-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_velocity_x1_evric.pdf}\label{fig_scenario2_x1_velocity_evric}}}}&
{\subfloat[The velocities of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the y-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_velocity_y1_evric.pdf}\label{fig_scenario2_y1_velocity_evric}}}}&
{\subfloat[The velocities of $\mathcal{P}_1$ and $\mathcal{V}_1$ along the z-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_velocity_z1_evric.pdf}\label{fig_scenario2_z1_velocity_evric}}}}\\
{\subfloat[The velocities of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the x-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_velocity_x2_evric.pdf}\label{fig_scenario2_x2_velocity_evric}}}}&
{\subfloat[The velocities of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the y-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_velocity_y2_evric.pdf}\label{fig_scenario2_y2_velocity_evric}}}}&
{\subfloat[The velocities of $\mathcal{P}_2$ and $\mathcal{V}_2$ along the z-axis.]{\scalebox{0.35}{\includegraphics[width=1.0\textwidth]{scenario2_velocity_z2_evric.pdf}\label{fig_scenario2_z2_velocity_evric}}}}
\end{array}
$
\caption{Scenario 2. The velocities of $\mathcal{P}_i$ and $\mathcal{V}_i$, for $i=1,2$, along the $x-, y-, z-$ axes. The red curves show the velocities of the point-mass robots while the blue curves show the velocities of the mobile landing platforms.}
\label{fig:scenario2_different_velocity_views_evric}
\end{figure*}
%%-----------------------------------------------------------------------------


%%-----------------------------------------------------------------------------
\begin{figure*}
$
\centering
\begin{array}{cc}
{\subfloat[Lyapunov function.]{\scalebox{0.47}{\includegraphics[width=1.0\textwidth]{scenario2_lyapunov_evric.pdf}\label{scenario2_lyapunov_evric}}}}&
{\subfloat[Lyapunov function time derivative.]{\scalebox{0.47}{\includegraphics[width=1.0\textwidth]{scenario2_lyapunov_derivative_evric.pdf}\label{scenario2_lyapunov_derivative_evric}}}}
\end{array}
$
\caption{Scenario 2. The behaviour of the Lyapunov function and its time derivative.}
\label{fig:scenario2_lyapunov_and_time_derivative_evric}
\end{figure*}
%%-----------------------------------------------------------------------------
\section{Conclusion}\label{section_conclusion_evric}
In this paper, we present an autonomous system for landing point-mass robots on car-like mobile landing platforms in $3$-dimensional space using LbCS. Herein, we have designed the nonlinear, time-invariant acceleration-based control algorithm for the AVs, which was extracted using the LbCS. Fixed and moving obstacles were included in the dynamic environment under consideration, which the point-mass robots and the mobile landing platforms avoided while in motion to their respective targets. The control laws take into consideration cylindrical obstacles that mimic buildings in a city-like environment. In addition, the MDT has been used to derive the mathematical functions related to cylindrical obstacles and their successful avoidance. To demonstrate the stability of the dynamic model, the direct Lyapunov approach is utilized. Computer simulations of real-world situations were used to demonstrate the effectiveness, robustness, and advantages of the control scheme. Future work will involve experimentally evaluating this strategy on an actual prototype.
%The V2V, V2I and V2X communications will be utilized for motion planning and control of the vehicular systems that would ensure collision-free navigation.



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